# Propability of finding a particle outside the classical limit

• Noctisdark
In summary, the conversation discusses the task of calculating the probability of finding a particle outside the classical limit for a quantum harmonic oscillator in its ground state. The relevant equations for this problem include the wavefunction and ground state energy of the oscillator. The individual attempting the solution is unsure of how to proceed and has considered the possibility of tunnelling. However, they are stuck at the point where they need to equate the energy to the potential. The conversation also includes a question about the validity of an equation and a clarification about the range of the quantum oscillator's position.
Noctisdark

## Homework Statement

I'm asked to calculate the propability of finding a particle outside the classical limit for a quantum harmonic oscillator in it's ground state

## Homework Equations

Ψ0(x) = a*emωx2/2hbar
When a = (mω/πhbar)1/4
The ground state energy E0 =hbar*ω2/2

## The Attempt at a Solution

I actually don't know what to do, I've tried to consider it as tunelling by finding when the energy is equal to the potential
E0 = hbar*ω2/2 = mω2x2, and stuck there !
And thank for your assistance

You might start by finding the amplitude, ##A##, of oscillation for a classical oscillator with energy E0. The quantum oscillator is "outside the classical limit" when x is outside the range ##-A\leq x \leq A##.

Noctisdark said:
E0 = hbar*ω2/2 = mω2x2

I don't understand this equation. Why is ##\omega## squared in the middle expression. I also don't understand the meaning of the expression on the right.

I think you have a typo in the expression for the wavefunction. The argument of the exponential should be negative.

Last edited:

## 1. What is the classical limit in particle physics?

The classical limit in particle physics refers to the boundary between classical mechanics and quantum mechanics. It is the point at which the behavior of a particle can no longer be accurately described by classical physics and requires the use of quantum mechanics.

## 2. How is the probability of finding a particle outside the classical limit calculated?

The probability of finding a particle outside the classical limit is calculated using the Schrödinger equation, which takes into account the particle's wave function, potential energy, and other factors. This equation is solved to determine the probability of finding the particle in a particular location.

## 3. What factors affect the probability of finding a particle outside the classical limit?

The probability of finding a particle outside the classical limit is affected by a variety of factors, including the particle's energy, location, and the strength of the potential barrier it is encountering. Other factors, such as the particle's spin and angular momentum, may also play a role.

## 4. Can a particle ever be found outside the classical limit?

It is possible for a particle to be found outside the classical limit, but the probability of this occurring decreases as the energy of the particle increases. At extremely high energies, the probability becomes so small that it is considered negligible.

## 5. Why is the study of particles outside the classical limit important?

The study of particles outside the classical limit is important because it allows us to better understand the fundamental nature of matter and the behavior of particles at a quantum level. It also has practical applications in fields such as quantum computing and nanotechnology.

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